This problem has been actively investigated in several settings.
[1]} first studied the upper bound on an operator norm of the deviation.
A representative topic is the case in which a matrix \(M_\ell \)  is high-dimensional.
[2]}, [3]}, [4]}, [5]} derived an upper bound that does not depend on the dimensionality, referred to as a dimension-free upper bound, using the notion of the effective rank of matrices.
Especially, [6]} studied an infinite-dimensional version of the problem.
This allows for the estimation of high-dimensional matrices without assuming sparsity [7]} or the shape of the distribution [8]}, [9]}, [10]}.
[11]} studied an asymptotically exact risk of the problem.
[12]} developed a bootstrap method for high-dimensional operators, and also studied a dimension-free bound in this setting.
For the case where the distribution of a matrix \(M_\ell \)  has a heavy-tail, [13]} derived a dimension-free upper bound to clarify how the tail property affects the bound.
[14]}, [15]} further improved the tightness of the upper bound for heavy-tailed distributions.
For a dependent setting of a sequence of matrices, [16]} derived a bound on the expected value of the norm of this perturbation, by following
[17]}.
